On Esscher Transforms in Discrete Finance Models

Abstract

The object of our study is where each S n is a m -dimensional stochastic (real valued) vector, i.e. denned on a probability space (Ω, , P) and adapted to a filtration ( n ) 0≤n≤N with 0 being the σ-algebra consisting of all null sets and their complements. In this paper we interpret as the value of some financial asset k at time n . Remark: If the asset generates dividends or coupon payments, think of as to include these payments (cum dividend process). Think of dividends as being reinvested immediately at the ex-dividend price. Definition 1 (a) A sequence of random vectors where is called a trading strategy. Since our time horizon ends at time N we must always have ϑ N ≡ 0 . The interpretation is obvious: stands for the number of shares of asset k you hold in the time interval [n,n + 1). You must choose ϑ n at time n . (b) The sequence of random variables where S n stands for the payment stream generated by ϑ (set ϑ −1 ≡ 0 ).